Wiley: Numerical Linear Algebra with Applications: Table of Contents

On Approximate Block Preconditioning for Discretized Optimal Control Problems Constrained with Space‐Fractional Diffusion Equations

Yu‐Hong Ran, Jia‐Ni Song — Mon, 08 Jun 2026 22:07:06 -0700

ABSTRACT

For a class of optimal control problems constrained with certain space-fractional diffusion equations, by making use of the right rectangular rule for the cost function and the implicit finite difference scheme with the shifted Grünwald formula for the constraint equation along with Lagrange multiplier approach, we obtain specially structured block two-by-two linear systems. We construct the circulant-based and τ$$ \tau $$-matrix-based approximate block preconditioning matrices for the coefficient matrices of the discrete linear systems and analyze spectral properties of the corresponding preconditioned matrices. Theoretical results indicate that except for a small number of outliers the eigenvalues of the preconditioned matrices are clustered around 1. Numerical experiments show that these structured preconditioners can significantly improve the convergence behavior of the Krylov subspace methods.

Row‐Aware Randomized SVD With Applications

Davide Palitta, Sascha Portaro — Mon, 08 Jun 2026 20:08:39 -0700

ABSTRACT

The randomized singular value decomposition proposed in [28] has certainly become one of the most well-established randomization-based algorithms in numerical linear algebra. The key ingredient of the entire procedure is the computation of a subspace which is close to the column space of the target matrix A∈ℝm×n$$ \mathbf{A}\in {\mathbb{R}}^{m\times n} $$, m≥n$$ m\ge n $$, up to a certain probabilistic confidence. In this paper, we employ a modification to the standard randomized SVD procedure, which leads, in general, to better approximations to Range(A)$$ \mathrm{Range}\left(\mathbf{A}\right) $$ at the same computational cost. To this end, we explicitly construct information from the row space of A$$ \mathbf{A} $$, enhancing the quality of the approximation for which we derive novel error bounds that improve over existing results. We also observe that very few pieces of information from Range(AT)$$ \mathrm{Range}\left({\mathbf{A}}^T\right) $$ may be necessary. We thus design a variant of this algorithm equipped with a subsampling step, which increases the efficiency of the procedure while often attaining competitive accuracy records. Our findings are supported by both theoretical analysis and numerical results.

Gram Decay and Intrinsic Dimensions of Krylov Subspaces

Stephen J. Thomas — Thu, 04 Jun 2026 02:22:45 -0700

ABSTRACT

Krylov subspace methods solve large sparse linear systems Ax=b$$ Ax=b $$ by building a sequence of polynomial approximations to A−1b$$ {A}^{-1}b $$ from successive matrix-vector products. In finite precision, the number of numerically independent directions that can be extracted from this sequence is bounded by the intrinsic information dimension kinfo$$ {k}_{\mathrm{info}} $$, defined as the index at which the Krylov basis matrix becomes numerically rank-deficient: σk+1(Vk+1)≤δσ1(Vk+1)$$ {\sigma}_{k+1}\left({V}_{k+1}\right)\le \delta \kern0.3em {\sigma}_1\left({V}_{k+1}\right) $$, where δ=O(εmachk2)$$ \delta =O\left({\varepsilon}_{\mathrm{mach}}{k}^2\right) $$ is the accumulated finite-precision noise floor. For sequences of elliptic PDE discretizations with C2$$ {C}^2 $$ spectral density, the normalized Gram matrix of any Chebyshev basis block decays off-diagonal at a rate Cρ/(i−j)2$$ {C}_{\rho }/{\left(i-j\right)}^2 $$, established via two integrations by parts applied to the Chebyshev moment integral; this gives kinfo≈κ$$ {k}_{\mathrm{info}}\approx \sqrt{\kappa } $$ (monomial basis) or ≈3κ$$ \approx 3\sqrt{\kappa } $$ (Chebyshev basis), where κ$$ \kappa $$ is the spectral condition number. Two consequences follow directly from this bound. First, any restarted Krylov method requires restart length m≥mmin≈2$$ m\ge {m}_{\mathrm{min}}\approx 2 $$–3κ$$ 3\sqrt{\kappa } $$ to reproduce unrestarted convergence; the common choices m=30$$ m=30 $$ or m=50$$ m=50 $$ produce 8–12.5×$$ 12.5\times $$ overhead or practical stagnation for κ=104$$ \kappa =1{0}^4 $$. Second, s$$ s $$-step GCR with Chebyshev basis and Forward Gauss–Seidel Gram solve requires only m=O(1)$$ m=O(1) $$ blocks of inter-block orthogonalization history, achieving O(msn)$$ O(msn) $$ storage and O(m/s)$$ O\left(m/s\right) $$ global synchronizations per iteration independent of iteration count; the 20–50×$$ 50\times $$ gap between this truncation depth and the restart minimum is explained by the same decay rate.

Highly Parallel Schwarz Preconditioning for Space‐Time Implicit Runge–Kutta Method for Parabolic Problems

Jing‐Yuan Wang, Shishun Li, Xiao‐Chuan Cai — Sun, 31 May 2026 23:02:29 -0700

ABSTRACT

The classical implicit Runge-Kuta (IRK) method is a powerful method with some desirable accuracy and stability properties but is rarely used in practical applications because of its high computational cost and difficulties in preconditioning. In this paper, we introduce a space-time coupled IRK scheme that offers high-order accuracy and stability, and also has a high degree of parallelism in both space and time when used with two-level tensor-structure-preserving overlapping Schwarz preconditioners. The convergence of the proposed method is studied numerically and we show that the convergence rate depends only mildly on the mesh size, the time step size, the number of processors, and the window size. We compare the parallel performance of the proposed method with the classical method in terms of the strong scalability, and the window-size-scaled weak scalability. The numerical results indicate that the proposed method outperforms the classical method when the number of processors is large and the space-only parallelization of the classical method is a limiting factor. Moreover, in terms of the total compute time, we show numerically that higher order space-time IRK outperforms the lower order space-time IRK when a suitable window size is chosen for solving the problem in the entire space-time domain with similar accuracy.

An Augmented Lagrangian Preconditioner for Navier–Stokes Equations With Runge–Kutta in Time

Santolo Leveque, Yunhui He, Maxim Olshanskii — Wed, 27 May 2026 04:02:12 -0700

ABSTRACT

We consider an implicit Runge–Kutta method for the numerical time integration of the nonstationary incompressible Navier–Stokes equations. This yields a sequence of nonlinear problems to be solved for the stages of the Runge–Kutta method. The resulting nonlinear system of differential equations is discretized using a finite element method. To compute a numerical approximation of the stages at each time step, we employ Newton's method, which requires the solution of a large and sparse generalized saddle-point problem at each nonlinear iteration. We devise an augmented Lagrangian preconditioner within the flexible GMRES method for solving the Newton systems at each time step. The preconditioner can be applied inexactly with the help of a multigrid routine. We present numerical evidence of the robustness and efficiency of the proposed strategy for different values of the viscosity, mesh size, time step, and number of stages of the Runge–Kutta method.

Chebyshev Smoothing With Adaptive Block‐FSAI Preconditioners for the Multilevel Solution of Higher‐Order Problems With the Partition of Unity Method

Pablo Jiménez Recio, Marc Alexander Schweitzer — Mon, 25 May 2026 22:01:41 -0700

ABSTRACT

In this paper, we assess the performance of adaptive and nested factorized sparse approximate inverses as smoothers in multilevel V-cycles, when smoothing is performed following the Chebyshev iteration of the fourth kind, for the efficient solution of linear systems arising from a conforming discretization of higher-order partial differential equations via the partition of unity method (PUM). To this end, we consider the (anisotropic) biharmonic and triharmonic equations in two and three dimensions and discretize these problems with a 𝒞2- or 𝒞3-regular PUM and polynomial exactness up to degree 5. We adapt existing adaptive algorithms for the construction of sparse approximate inverses to the native block structure of matrices arising in the PUM. Additionally, we also present a simplified formulation of the Chebyshev iteration of the fourth kind.

A Factorized Column Sparse Approach to the CP Rank Regularized Tensor Optimization Problem

Kaixin Gao, Yang Xu — Tue, 19 May 2026 16:38:56 -0700

ABSTRACT

Tensor optimization problems with rank regularization have attracted significant attentions in recent years due to their extensive applications in various fields. In this paper, we focus on the CANDECOMP/PARAFAC (CP) rank regularized tensor optimization problem and equivalently transform it into a column sparse regularized problem, that is, the problem with a regularization to describe the column sparsity of CP factor matrices. Moreover, we study the relationships between the original problem and the transformed problem in the sense of global and local minimizers. To solve the nonconvex and nonsmooth transformed problem, we design an inertial block coordinate descent (iBCD) algorithm and establish its global convergence. Finally, we apply the proposed iBCD algorithm to solve tensor CP decomposition and low-CP-rank tensor completion problems. Numerical experiments on both synthetic data and real-world images validate the promising performance of our proposed method compared with several excellent methods for solving the CP rank regularized tensor optimization problem.

Modified Greedy Quasi Block Coordinate Descent Method for Solving Linear Least‐Squares Problems

Fatemeh P. A. Beik, Khalide Jbilou — Mon, 11 May 2026 02:49:48 -0700

ABSTRACT

Recently, a greedy quasi-block coordinate descent (GQBCD) method was proposed for solving overdetermined linear least-squares problems as published in [Applied Mathematics Letters, 171(2025), 109675]. The GQBCD method is based on a random partition and the working block is determined by a greedy strategy at each step. In this paper, we introduce a modified version of the greedy quasi block coordinate descent method, referred to as MGQBCD, and provide a theoretical analysis of its convergence properties. The proposed approach enhances the original GQBCD framework by generating updated approximations through error minimization over an expanded subspace formed by two working blocks. Numerical experiments confirm both the feasibility and the improved performance of the MGQBCD method compared to the original GQBCD, particularly in terms of accuracy and convergence behavior.

A Preconditioned Majorization‐Minimization Method for ℓ2$$ {\ell}^2 $$‐ℓq$$ {\ell}^q $$ Minimization

A. Buccini, M. Donatelli, M. Ratto, L. Reichel — Wed, 06 May 2026 21:01:19 -0700

ABSTRACT

The need to minimize a linear combination of an expression that involves an ℓq$$ {\ell}^q $$-norm of a linear transformation of the computed solution and the ℓ2$$ {\ell}^2 $$-norm of the residual error arises in image restoration as well as in statistics. A solution method that is suited for large-scale problems is the iterative maximization-minimization method, which determines an approximate solution in a generalized Krylov subspace. This paper describes a preconditioner for accelerating the convergence of the iterative method. The design of the preconditioner is inspired by the iterated Tikhonov regularization method. Numerical examples in image deblurring and computerized tomography show that the preconditioner significantly reduces the required CPU time and gives computed solutions of about the same quality as the unpreconditioned iterative method.

Issue Information

Wed, 06 May 2026 20:59:44 -0700

Numerical Linear Algebra with Applications, Volume 33, Issue 3, June 2026.